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Section: New Results


Phase diagram for higher-order active contour models

Keywords : higher-order, active contour, phase diagram, stability, shape, prior.

Participants : Aymen El Ghoul, Saloua Bouatia, Peter Horvath, Ian Jermyn, Josiane Zerubia.

This work was performed as part of, and was partially funded by: ACI QuerySat [ ]; the INRIA STIC-Tunisia programme (Sup'Com Tunis); EU project MUSCLE [ ] and Egide PAI Balaton.

Higher-order active contours (HOACs) are a region modelling framework that allows the inclusion of sophisticated prior information about the geometry of the region modelled. For example, network shapes (used for road network extraction) and a `gas of circles' (used for tree crown extraction) can both be modelled in the HOAC framework. Varying the parameters of a model can give rise to widely varying behaviours. For example, the two models cited above differ only in the values of their parameters, yet the information they contain is quite different. This is illustrated in figure 24 , where gradient descent evolutions are shown, starting from a rounded square, for different parameter values. As can be seen, the square may evolve into a network shape or a gas of circles, or it may disappear. In order fully to understand these behaviours, we are constructing a `phase diagram' for the above models, i.e. the map between the parameter space and the nature of the local minima. Such a diagram will not only allow the selection of appropriate parameter values for a given application. It will also enable the exploration of regions of the phase diagram where the behaviour is unknown, and hence lead to models incorporating new types of prior information, as well as giving insight into the capabilities of the HOAC framework in general.

To construct the diagram, we study the stability of shapes under the energy functional. The energy is expanded in a functional Taylor series around the shape being analysed, and the conditions defining a local minimum are imposed. This leads to multiple constraints on the parameters, and defines a region in the phase diagram where this shape is stable. Similarly, in parts of the phase diagram where the shape is unstable, its principal instabilities can be described. So far, this analysis has been performed for a circle, corresponding to the gas of circles model, and for a long bar, corresponding to the network model. We are now combining the results of these analyses to produce the phase diagram.

Figure 24. Gradient descent evolutions under a HOAC energy for different parameter values. Top row: leading to a disappearing region; middle row: leading to a network shape; bottom row: leading to a gas of circles.


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